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Blake, I'm currently writing a paper on pointfree topology (closely related to mereotopology) and Contact Manifolds, which I'm sure you'll find fascinating. It's mostly written under the language of enriched categories and Monoids (which are Petri Nets of sorts). In essence, they should provide us with a greater understanding of Process Calculi, and how an 'automata group' can be studied under a 'complex' (although the term is symplectic, but we're focusing on the complement of symplectic, which is 'contact') manifold. In essence, our findings should pave the way to new insights on solving problems in 4-Manifolds. Such as the isotopy problem and counting pseudo-holomorphic curves, but combinatorially. I have been in correspondence with a couple of professors.
Here's a clue:
| quote: | Five silent philosophers sit at a table around a bowl of spaghetti. A fork is placed between each pair of adjacent philosophers.
Each philosopher must alternately think and eat. Eating is not limited by the amount of spaghetti left: assume an infinite supply. However, a philosopher can only eat while holding both the fork to the left and the fork to the right (an alternative problem formulation uses rice and chopsticks instead of spaghetti and forks).
Each philosopher can pick up an adjacent fork, when available, and put it down, when holding it. These are separate actions: forks must be picked up and put down one by one.
The problem is how to design a discipline of behavior (a concurrent algorithm) such that each philosopher won't starve, i.e. can forever continue to alternate between eating and thinking. |
Trace monoids are commonly used to model concurrent computation, forming the foundation for process calculi. They are the object of study in trace theory. The utility of trace monoids comes from the fact that they are isomorphic to the monoid of dependency graphs; thus allowing algebraic techniques to be applied to graphs, and vice-versa.
| quote: | Sheaves, Stacks, and...um, Space...kind of:
The basic idea of sheaves
To understand what a sheaf is, recall which aspect of the notion of topological space was relevant in the above examples ℬnU(1): these spaces were entirely characterized by how one can map other spaces into them:
X↦[X,ℬnU(1)].
This is a general strategy that one can adopt: suppose I dream up a space but don’t tell you which one it is, but I give you hints: for each other space U that you can dream up, I tell you how you can probe my space by mapping your space U into it.
Let’s now call the space which I dreamed up X, the generic symbol for spaces. For every space U that you come up with, I do some secret computation and then present you with the result: I hand you a set, let’s call it X(U), and tell you that this is the set of ways that U can be mapped into X:
X(U)={ways to map U into some hypothetical X}
=: {probes of X by U}.
will you be able, in general, to guess my space X from this information U↦X(U)? No. So, to be fair, I should provide a bit more information: what you need to know to actually get a feeling for what my space X is like is an idea about how different ways of probing my space relate to each other.
Okay, so we’ll do this: for every pair U and V of test spaces that you come up with, and for every way ϕ:U→V of mapping these into each other, I inform you not only about the set of ways X(U)={U-probes of X} that U can be mapped into my secret space X, and the set of ways X(V)={V-probes of X} that V can be mapped into my secret space, but I also tell you how these are related when you first map V into X by a map (p:V→X)∈X(V) and then map U into X by first mapping it to V: U
−→fV−→pX.
This transforms every element in X(V) into an element in X(U). To be fair, I should tell you at least what this transformation is!
So I’ll do that: for every map of test spaces f:U→V that you hand me, I return you a map of sets that I denote X(f):X(V)→X(U), which indicates how V-probes of X turn into U-probes when you precompose them with f:U→V.
Do you need still more information to guess my space X?
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http://ncatlab.org/nlab/show/motivation+for+sheaves%2C+cohomology+and+
higher+stacks
Good luck!
Last edited by Lagrangian on May-01-2012 at 23:56
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Long time, no see, man!
